College Trigonometry, 2011a

11.7 Polar Form of Complex Numbers 997
The following theorem summarizes the advantages of working with complex numbers in polar form.
Theorem 11.16. Products, Powers and Quotients Complex Numbers in Polar Form:
Suppose z and w are complex numbers with polar forms z = |z|cis(α) andw = |w|cis(β). Then
ˆ Product Rule: zw = |z||w|cis(α + β)
ˆ Power Rule (DeMoivre’s Theorem) : z n = |z| n cis(nθ) for every natural number n
ˆ Quotient Rule:
z
w = |z| cis(α − β), provided |w| ≠0
|w|
The proof of Theorem 11.16 requires a healthy mix of definition, arithmetic and identities. We first
start with the product rule.
zw = [|z|cis(α)] [|w|cis(β)]
= |z||w| [cos(α)+i sin(α)] [cos(β)+i sin(β)]
We now focus on the quantity in brackets on the right hand side of the equation.
[cos(α)+i sin(α)] [cos(β)+i sin(β)] = cos(α) cos(β)+i cos(α)sin(β)
+ i sin(α) cos(β)+i 2 sin(α)sin(β)
= cos(α) cos(β)+i 2 sin(α)sin(β) Rearranging terms
+ i sin(α) cos(β)+i cos(α)sin(β)
= (cos(α) cos(β) − sin(α)sin(β)) Since i 2 = −1
+ i (sin(α) cos(β) + cos(α)sin(β)) Factor out i
= cos(α + β)+i sin(α + β) Sum identities
= cis(α + β) Definition of ‘cis’
Putting this together with our earlier work, we get zw = |z||w|cis(α + β), as required.
Moving right along, we next take aim at the Power Rule, better known as DeMoivre’s Theorem. 9
We proceed by induction on n. LetP (n) be the sentence z n = |z| n cis(nθ). Then P (1) is true, since
z 1 = z = |z|cis(θ) =|z| 1 cis(1 · θ). We now assume P (k) is true, that is, we assume z k = |z| k cis(kθ)
for some k ≥ 1. Our goal is to show that P (k + 1) is true, or that z k+1 = |z| k+1 cis((k +1)θ). We
have
z k+1 = z k z Properties of Exponents
= ( |z| k cis(kθ) ) (|z|cis(θ)) Induction Hypothesis
= ( |z| k |z| ) cis(kθ + θ) Product Rule
= |z| k+1 cis((k +1)θ)
9 Compare this proof with the proof of the Power Rule in Theorem 11.14.